We present recent work on treecode algorithms for computing integrals that arise in particle simulations. Our approach uses a Cartesian Taylor series for the far-field expansion and is suitable for a variety of kernels including multiquadric radial basis functions and the screened Coulomb potential. A sample of results will be presented for vortex sheet motion in 3D fluid flow and boundary integral simulations of the Poisson-Boltzmann equation for solvated biomolecules.